In order to address the financial network problem, previous work employed D-Wave quantum annealer, which attracts significant attention as a metaheuristic for combinatorial optimisation problems [3] . Since Ising machines such as D-Wave quantum annealer require binary formulations of problems such as Quadratic Unconstrained Binary Optimisation (QUBO), real numbers need to be encoded as binary variables. In addition, the Heaviside functions included in the original problem formula must also be transformed to polynomials by Legendre expansion, and order-reduction methods were applied to limit the highest-order terms to second order. This process results in an increase in variable numbers, which often worsens solution accuracy. Although this approach allowed us to obtain solutions for small-size problems, the hardware limitations of the D-Wave quantum annealer restricted its ability to scale to larger problem sizes. Additionally, approximation errors originating from the Legendre expansion could prevent us from obtaining high-accuracy solutions.
In our approach, we developed a new encoding method to express real numbers utilising sign bits. This technique enables the substitution of the Heaviside functions of the original formula, without any approximations. Using our proposed method, we obtained a Polynomial Unconstrained Binary Optimisation (PUBO), whose highest order is three. As the Ising machine, we employed a Simulated Bifurcation Machine (SBM), whose PUBO solver can handle up to fourth-order terms without order reductions. Finally, our proposed approach outputs high-accuracy results for certain problems, including larger problem instances than the previous work.
